# Calculate a thermal expansion pitch change

#### PianoTek

Can anyone please help me figure out how to calculate the change in pitch [in cents] for a piano wire with a given change in temperature? I am not really sure where to start.

I assume that I will need to first figure out the thermal expansion of the wire, but how do I then take that information and calculate the new tension and resulting pitch?

Also, how long does it take something like a steel piano wire to change temperature with a change in ambient temperature (e.g., if the room temperature changes from 68F to 70F, how long does it take for the string to change temperature)?

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#### Mapes

Homework Helper
Gold Member
Hi PianoTek, welcome to PF. To address your second question first, an exposed piano wire will match a temperature change within one or a few seconds. If the wire is inside a chamber (in an upright, say), the lag may be longer because the air inside will match the chamber temperature, and the chamber's larger mass will be slower to change temperature (minutes instead of seconds).

You can combine a few equations to estimate the change in pitch with temperature. I'm sure you already know the equation relating wire tension to pitch. To this you should add (1) Hooke's Law

$$F=\frac{l_s-l_u}{l_u}EA$$

where F is the wire tension (units of force), $l_s$ is the stretched wire length, $l_u$ is the unstretched (original) wire length, E is the wire stiffness (Young's modulus) and A is the wire area; and (2) the thermal expansion equations

$$l_{s}=l_{s,0}(1+\alpha_f \Delta T)[/itex] [tex]l_{u}=l_{u,0}(1+\alpha_s \Delta T)[/itex] which describe how the frame material and the string material elongate or contract according to their individual thermal expansion coefficient $\alpha$ and the temperature change $\Delta T$. The zero subscripts correspond to lengths at a reference temperature from which $\Delta T$ is measured. So for change in tension we have [tex]\Delta F=\left(\frac{l_{s,0}(1+\alpha_f \Delta T)-l_{u,0}(1+\alpha_s \Delta T)}{l_{u,0}(1+\alpha_s \Delta T)}-\frac{l_{s,0}-l_{u,0}}{l_{u,0}}\right)EA$$

$$\Delta F\approx \left(\frac{(l_{s,0}\alpha_f-l_{u,0}\alpha_s) \Delta T}{l_{u,0}}\right)EA$$

$$\Delta F\approx (\alpha_f-\alpha_s) EA\Delta T$$

where I've made successive approximations by first assuming that $\alpha_s \Delta T$ is small and then by assuming that $l_{s,0}\approx l_{u,0}$. You can see that if the frame expands more (less) than the string, the pitch will rise (fall). Does this help?

#### rbj

a good-looking webpage for waves on a string might be:
http://hyperphysics.phy-astr.gsu.edu/Hbase/waves/string.html

and, for pitch in cents (remember there are 1200 cents per octave), the relationship between a change in frequency and the corresponding change pitch in cents is:

$$c_1 - c_0 = 1200 \log_2 \left( \frac{f_1}{f_0} \right)$$

the other important relationship is that that a long metal rod (what the string would be if it wasn't stretched) will lengthen or shorten roughly in direct proportion to the absolute temperature (Kelvin). an important unknown that Mapes pointed out is, when the temperature changes, how does the piano frame expand or contract relative to the string length? if they expand or contract precisely together (for small temperature changes), i am not sure that the string tension would change at all.

#### PianoTek

Thank you so much for giving me a place to start. I am ashamed to say that I have no experience in the area of physics (even with multiple degrees in music) . . . so please bare with me; a lot of this looks like an alien language to me right now!

Let me give some background information: I am a concert piano technician that is really sensitive about the temperature when I tune. I know from my own experience that temperature has a huge impact on the quality of the tuning. What I am trying to do now is mathematically understand what is going on and to be able to better predict what would happen with any given change. This might allow me to better compensate for those odd situations when they do occur.

I know for sure that small changes in the room temperature will quickly cause changes in the tuning. Sometime I will see temperature changes [maybe in the 4-7F range] that will result in a pitch change somewhere in the 3-5 cent range. I usually notice this when I have finished the top section of the piano: I then go back to recheck the middle section and notice that it has shifted in one direction or the other because of temperature.

I am guessing that the pitch change might be dependant upon the length of the string (i.e., a string that is twice as long might have twice the amount of change). I think that the problem seems to compound itself as I near the end of the tenor section, and is less significant in the highest section of the piano. I should note that there are many other issues that could easily explain this experience though . . .

I know for sure that cold temperatures make the piano go sharp, and warmer temperatures make the piano go flat. This is exactly opposite of wind players.

I am guessing that the iron frame can play a role in temperature/pitch change. But I think that because of its mass, it takes much longer to change temperature, and when it does, it should work in the opposite direction (i.e., when the plate expands because of temperature, the pitch should go up).

I do have experience with a concert piano that had been left in the truck overnight in really cold weather. When it was delivered in the morning for the concert, the plate was REALLY cold to the touch. It took many hours for the plate to warm up . . . and I can honestly say that it was the worst tuning I ever did. It took nearly three hours before I just gave up. I started the tuning immediately with a cold room, cold strings, cold case, and a cold plate. As lights turned on and things started to warm up, the tuning seemed to go all over the place . . . it was a nightmare.

The other [probably very small] part of the equation is how changes in temperature might affect the pitch (i.e., like the plate, this factor would also be contrary to my experience with warm air making the piano go flat). So, if I understand correctly, the speed of sound increases with warmer temperatures, but does this mean that the pitch sounds slightly higher with an increase in temperature?

I have worked with the cent/frequency equation before, so I am pretty comfortable with that. But can you help me get started plugging numbers in to the other equation?

Another issue is that the soundboard is crowned. Would small changes in temperature affect the expansion of the wood enough to take this into consideration? If the wood expanded, the soundboard would go up and increase the pitch (again, this is opposite to my experience).

Any guidance would be GREATLY appreciated!

#### Mapes

Homework Helper
Gold Member
PianoTek, this is really interesting. A few thoughts:

- Since resonant frequency decreases with increasing length, the temperature effect we're discussing should be less significant for longer strings. This is not what you observed, which is a little puzzling. As you said, however, there may be other effects at work.
- A quick internet search indicates that most steels have a higher coefficient of thermal expansion (CTE) than iron (but both are around 11 x 10-6 per °C). This is in agreement with a piano going flatter with increased temperature; the string expands more than the frame and consequently "relaxes." Here we're up against a great materials problem: the spring steel used in the piano string is carefully engineered to be extremely strong. The frame is relatively massive and is ideally as cheap as possible and easily cast. The need to use two different materials makes a CTE mismatch almost inevitable.
- The speed of sound in air shouldn't make any difference, if that's what you meant. The speed of sound in the steel string is directly related to the temperature effect we're calculating; if it's higher, the pitch increases, and vice versa.
- The soundboard also expands with temperature, likely even more than the metal. But am I correct in assuming that it bears little load compared to the iron? If it plays little part in keeping the strings tight, then it will also have little influence on the temperature effect.

Staff Emeritus
I know for sure that cold temperatures make the piano go sharp, and warmer temperatures make the piano go flat. This is exactly opposite of wind players.
That's because completely different physics is at work. For wind players, the key is the speed of sound in air. Essentially, a wind instrument is a constant wavelength instrument, but our ears respond to frequency. The same physics governs the change in voice when breathing helium: that too changes the speed of sound.

The key to the piano is to recognize that the frequency is related to the tension. $$f \approx \sqrt{T}}$$ if I recall correctly. Suppose properly tensioned a wire is stretched by 0.1%. Now raise it's temperature by ~10K, which causes it to lengthen by 0.01%. (Most materials have a coefficient of thermal expansion in that ballpark). So now it's only stretched by 0.09%, so it has 10% less tension, so the frequency is 5% less.

"Calculate a thermal expansion pitch change"

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